Problem: Let $f(x)=5\cdot2^x$. Find $f'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $5\cdot 2^x\ln(x)$ (Choice B) B $5\cdot 2^{x-1}$ (Choice C) C $5\cdot 2^x\ln(2)$ (Choice D) D $5\cdot 2^x\log_2(x)$
Solution: The expression for $f(x)$ includes an exponential term. Remember that the derivative of the general exponential term $a^x$ (where $a$ is any positive constant) is $\ln(a)\cdot a^x$. Put another way, $\dfrac{d}{dx}(a^x)=\ln(a)\cdot a^x$. $\begin{aligned} f'(x)&=\dfrac{d}{dx}(5\cdot2^x) \\\\ &=5\dfrac{d}{dx}(2^x) \\\\ &=5\cdot\ln(2)\cdot2^x \\\\ &=5\cdot2^x\ln(2) \end{aligned}$ In conclusion, $f'(x)=5\cdot2^x\ln(2)$.